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     There is an extensive literature on theoretical intervals and gens (ajnas) in Arabic music. But we dispose of less information on how the musician, its priest, play this music.


     The acoustical analysis of the music of several non western modal music, specially the Arabic, has shown the existence of  tetrachordal divisions which differ of the traditionally accepted. We center our attention on the ton, grater than the diatonic one, 9/8, which can be represented by the 8/7 ratio.


     In the following paragraphs we review the theoretical divisions of tetrachord, ajnas, which include this great tone, and contrast these ajnas with the results of our research, which allow us to recognize the trend of some interpreted music toward these interval and ajnas.


The Theory: Tetrachordal divisions and Consonant intervals.


     We consider that modal music repose on consonant intervals. These intervals are present in all maqams, the strings of instruments, the harmonics of wind instruments. Even maqams and ajnas with non simple tonic, as sikah, use consonants intervals as reference and support (i.e. rast‑naua).


     Therefore we shall confine our list to the tetrachordal divisions which uses consonant intervals. And we consider the more consonant the more simpler is the ratio of frequencies, and over all, those of the form (m+1)/m, being m an positive integer. This is the traditional approach, which we adopt.


     However these limitations the possibilities are very numerous. Usually, when dealing with European modal music additional restrictions are imposed: only quintal and tertian harmony are admitted, that is, only 2,3,5 are the prime numbers which are accepted as factors in the numbers which appear in the interval ratio. Thus 9/8 are accepted because 9/8 = 3*3/2*2,  but  not 11/10, because 11 is a prime number.


     But, for Arabic and other related music (possibly we could say, Islamic music) these restrictions are too narrow, as they cannot explain the 'quartertone intervals'. We admit therefore also the numbers 7, 11 and 13, which allow more flexible divisions. Indeed, the neutral tierce is well represented by the 11/10,  12/11, 13/12 or 14/13 intervals.


     Here we shall see only those ajnas which includes the great ton or 'tonus maximums', 8/7, as one at least of its three intervals. As the complement to the quart, 4/3, is  7/6 ( = (4/3)/(8/7) ), we look for two ratio which product is 7/6: the possibilities are:              



       INTERVALS RATIO                  INTERVALS in CENTS

1.   8/7 .  8/7  . 49/48    =   4/3     231 + 231 +  38  =   498 

2.   8/7 .  9/8  . 28/27    =   4/3     231 + 204 +  63  =   498

3.   8/7 . 10/9  . 21/20   =   4/3     231 + 182 +  85  =   498

4.   8/7 . 13/12 . 14/13  =   4/3     231 + 139 + 128  =  498


5.   8/7 . 25/24 . 28/25  =   4/3     231 + 196 +  71   =  498

     But we see that the fifth division and other similar ones are no longer of the form (m+1)/m. And the first presents an interval of 38 cents which is too little to be considered an melodic interval. Only three ajnas remain, then.


     These ajnas are well‑known from the antiquity. The Greeks described and Ptolomeus (180 B.C.), called them 'middle diatonic' for our second division and  'smoothed diatonic' for the third. See Gevaert [1], pp 315.  Also the Arabic theoretician treated them. Al‑Farabi quote and describe the first 3 ajnas, and even recommend the first and second  for the execution on the Tanbur of Baghdad; see d'Erlanger [2], tome 2, notes 4,5 to Al‑Farabi book, page. 248. Ibn‑Sina too include  these ajnas, as Farabi does, within the class 'strong'.


     If we change now the order of these intervals we find 6 combinations for each jins. We will name those combinations the 'rast type', the 'hiyas type' and the 'kurdi type' according the position of the 8/7 interval in first, second or third positions, respectively. The position of the middle interval provide us with three other types. We have thus six form:

   TYPE          ORDER of INTERVALS              ASPECT


  rast             greater  middle    smaller  ...................

  nahauand   greater  smaller   middle   ...................

  hiyasA       middle   greater   smaller  ...................

  baiati         middle   smaller   greater  ...................

  hiyasB       smaller  greater   middle   ...................

  kurdi          smaller  middle    greater  ...................


      According to the actual used tetrachord division we will name rast‑2 the  231‑204‑63 division or  baiati‑3  the 182‑85‑231 division.


Experimental Results

     Applying our method of statistical analysis of intervals, described in Sanchez [3], we have found melodic intervals which are functionally  whole tones, that is, they are perceptively whole tones for opposition to semitones, neutral tones and augmented tones.


     But our measurements shows without any doubt, that, in many occasions, their size is larger: we  find often values of 224, 232 and 240 cents, nearer of the 8/7 interval (231 cents), than of the diatonic whole tone, 9/8 (204 cents).


     And it is surprising to find them in instrumental and vocal music, in Indian, Andalusian, Arabian, Turkish and Spanish (cante hondo) music. And that in the same places of scale, the first interval of a rast type tetrachord, and the second interval of a hiyas1. Lets see some results:



     We performed extensive measurements on music from different cultures, but having some current or historical link with arabic or islamic culture and music. We include in that Arabic countrys, from middle East to Maroc, Spain, in the West. And in the East, from Persian to India.


     Here are some results, with information about country, mname of player(s), musical instrument, maqam or similar concept of fragment, tetrachord inside of the scale of the maqam, intervals found in the analysis, type of the tetrachor according our proposal, and order of division of tetrachord as listed before. The graphics of some analysis can be also seen.

Country Musician Instrument Maqam Tetrachord Intervals  Type Div.


Liban  Ba'yun        tanbur    rast             1     236  147  141   rast ‑4

Liban  Ba'yun        tanbur    rast             2     222  141  128   rast ‑3

Spain  Menese      voice..   martinete     1     243  165   98    rast ‑3

India  Banerjee     sitar       charukesi     1     233  164   102   rast ‑3

Iraq   M.Bachir     'ud         nahauand     2     242  182   75    rast ‑3

Maroc Rais orq.   voi&orq istihlal             223   164  120  rast ‑3

Turkey Öczimi      nai         rast               2     243  101  157   nahau‑3

Liban  Ba'yun        tanbur    rast             2     133  235  142   hiyaB‑4

Iraq   S.Chukur      'ud         baiati           1     134  146  224   kurdi‑4

 We found also these intervals in Javanese music, but out of the tetrachord boundaries.


      The analysis have been made on long fragments, in order to detect the 'trend' of the ear and finger of player. We have selected, naturally, only the results that ilustate our point of view: we do not affirm that these maqams and ajnas must be played with these intervals, only that they are sometimes played in this way by quality musicians. See a description of analysed records in appendice 1.



     How can we explain these intervals ?. In some cases they came out of the instrument: it is the case of the tanbur, which does not allow other "whole tones" that 182 cents (10/9) and 231 (8/7), due to the fixed frets (see Helmoltz [4] and Sanchez&Odeimi [7] ). We must remark that those frets and intervals are not arbitrary, they crather came from the tradition, and therefore agree with the musical system.


     In the case of the 'ud we can explain that for the positions of the fingers and/or for the tuning of strings. If we tune the main strings asiran‑dukah‑naua‑kirdan of the 'ud in the usual Pythagorean way, we have the 8/7 intervals between dukah, free string and rast on asiran string, by raising the dukah about one coma, until its seventh armonics harmonizes with kirdan. (we tried this tuning and the result was far of being chocking).


     But in vocal music we have no instrumental raisons: we must suppose that the ear and the spirit, in order to perceive the ajnas and maqamat differences, reacts in the same apparently crude way shown in the list of 'tetrachord types', by comparing the relatives sizes of intervals to order them as 'greater than' or 'smaller than'.


     Does it means that any interval produces the same effect than another belonging to the same category otr type?. Not at all, but we believe that each type represents a 'family'.


     These families can be recognized in practical music: for instance, Baiati presents a fist interval which varies greatly, between about 133 cents in the persian 'schur' until about 180 cents in turkish 'uchak', that is, between a 'big semitone' and a 'small minor tone'. And even within the arabic music, the range is no less of 135‑169 cents (see d'Erlanger [5] and Touma [6]).


     Inside of the family, subfamilies can be found according to the country, region and even individuals. But a link can be traced between these variations: the common type, described before.



     The apparition of intervals in the surrounding area of the 8/7 interval, or great tone, suggests that Pythagorean scale cannot explain all the actual intervals played in Arabic and other related musics. It seems to prove that other divisions of tetrachord are also in use nowadays in the ear of the musician. And lets remember once more that he is the door through music comes out.


Appendice 1


     The players and musical recordings analyzed and described here are:


Mohiuddin Ba'yun, tanbur. Liban.  Baidaphon Company B 082 703, Cairo, 1920. Reedited in CD by  Institut du Monde Arabe, vol 1, Ocora, Paris,1987. Rast taqsim extensively analyzed in Sanchez&Odeimi

[7]Menese, singing martinete, Spain. Private cassette with selections of non‑instrumental flamenco (hondo) music, made by D.Antonio Espinós, expert in this field. First quatrain. Martinete is of Rast type in first notes.

Nikhil Banerjee, sitar, India.

     Recordings of Ragas Bhairavi and Charukesi. Raga Charukesi  has first tetrachord of rast type.

Munir Bachir, 'ud, Iraq.

     Le 'Ud Classique Arabe. Harmonia Mundi, Paris 1984. Taqsim  nahauand.

Maroc. Musique clasique andalou‑mahgrebine.

     Orquestre de Fez, dirigée par Hajj Abdelkrim Raïs. Face B. Baytain (vocal solo) in nouba Istihlal (similar to Rast)  sung by Abdel Rahim Suiri. Ocora, HM 57.588 588. Paris, 1986

    Ustad Öczimi, turkish ney. Professor in Islamic University of  Konia, Turkey. Private recording of main maqamat specially  made for the author. Maqam rast.

    Salman Chukur, 'ud. Iraq. Iraq, craddle of Civilization, Ministry of Culture, Baghdad,  1982. First record (of three). Taqsim Baiati.






[1] GEVAERT 'Histoire et theorie de la musique de l'Antiquité, Olms Verlg.Hildesheim,1965.


[2] D'ERLANGER, R. La Musique Arabe. Tome II.Geuthner. Paris, 1935. notes 4,5 to Al‑Farabi book, pag. 248.


[3] SANCHEZ GONZALEZ,F.J. 'Analisis de intervalos y escalas modales'. Revista Española de Musicología, Madrid, 1989.


[4] HELMHOLTZ, H., On the Sensations of Tone,pp 517.Dover,New York,1954. Pag 517.


[5] D'ERLANGER, R.. La Musique Arabe. Tome V. Geuthner. Paris, 1949. pag.47.


[6] TOUMA, H.H. 'The maqam baiati in the arabian taqsim'.Arabiche buch. Berlin 1980.Pag.10


[7] SANCHEZ, F.J. &  ODEIMI,B. 'Etude sur le maqam dans un taqsim de tanbur'. In publication.



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